Physics 3330 Experiment #4 Fall 2013

Experiment #4 4.1

Operational Amplifiers and Negative Feedback

Purpose

This experiment shows how an operational amplifier (op-amp) with negative feedback can be

used to make an amplifier with many desirable properties, such as stable gain, high linearity, and

low output impedance. You will build both non-inverting and inverting voltage amplifiers using

an LF356 op-amp.

Introduction

The purpose of an amplifier is to increase the voltage level of a signal while preserving as

accurately as possible the original waveform. In the physical sciences, transducers are used to

convert basic physical quantities into electric signals, as shown in Figure 4.1. An amplifier is

usually needed to raise the small transducer voltage to a useful level.

A small voltage or current change at the input of the amplifier controls a much larger signal at

the input of whatever circuit or instrument the amplifier’s output is connected to. Measuring and

recording equipment typically requires input signals of 1 to 10 V. To meet such needs, a typical

laboratory amplifier might have the following characteristics:

1. Predictable and stable gain. The magnitude of the gain is equal to the ratio of the output

signal amplitude to the input signal amplitude.

2. Linear amplitude response, so that the output signal is directly proportional to the input

signal.

Transducer

Oscilloscope

Data Analysis System

Amplifier

Temperature, Magnetic Field, Radiation, Acceleration, etc.

DC Power

Volts to mVolts

Figure 4.1 Typical Laboratory Measurement

Volts

Experiment #4 4.2

3. According to the application, the frequency dependence of the gain might be a constant

independent of frequency up to the highest frequency component in the input signal

(wideband amplifier), or a sharply tuned resonance response if a particular frequency

must be picked out.

4. High input impedance and low output impedance are usually desirable. These

characteristics minimize changes of gain when the amplifier is connected to the input

transducer and to other instruments at the output.

5. Low noise is usually important. Every amplifier adds some random noise to the signals it

processes, and this noise often limits the sensitivity of an experiment. We will learn

about amplifier noise in Experiment 8.

Commercial laboratory amplifiers are readily available, but a general-purpose amplifier is

expensive (>$1000), and most of its features might be unneeded in a given application. Often, it

is preferable to design your own circuit using a cheap (<$1) op-amp chip.

Op-amps have many other circuit applications. They can be used to make filters, limiters,

rectifiers, oscillators, integrators, and other devices (see FC 12.9 – 12.15). To get an idea of the

variety of op-amps available, have a look at the National Semiconductor web site

(www.national.com). They currently make over 300 types, offering trade-offs between speed,

cost, power consumption, precision, etc. The op-amp is the most important building block of

analog electronics.

Suggested Readings

1. FC Sections 12.2 – 12.15. The basic rules of op-amp behavior and the most important

op-amp circuits. [Note while FC discussed basic characteristics of op amps in 12.2 it

does not have the I, II “Golden Rule” analysis that we will discuss in class.] Do not worry

right now about the transistor guts of op-amps. We will learn about transistors in

Experiments 7-8.

2. Horowitz and Hill, Chapter 4.

3. The IC Op-Amp Cookbook, by Walter Jung, 3rd

edition (Prentice Hall, 1997).

4. Diefenderfer Chapter 9

5. D.V. Bugg, Chapter 7, particularly Sections 7.1–7.4, 7.7, and 7.12.

Experiment #4 4.3

Theory

The two basic op-amp circuit configurations are shown in Figs. 4.2 and 4.3. Both circuits use

negative feedback, which means that a portion of the output signal is sent back to the negative

input of the op-amp. The op-amp itself has very high gain, but relatively poor gain stability and

linearity. When negative feedback is used, the circuit gain is greatly reduced, but it becomes

very stable, limited only by the temperature dependence of the resistors R and RF. At the same

time, linearity is improved and the output impedance decreases.

Both configurations are widely used because they have different advantages. Besides the fact

that the second circuit inverts the signal, the main differences are that the first circuit has a much

higher input impedance, while the second has lower distortion because both inputs remain very

close to ground.

The gain or transfer function A of the op-amp alone is given by

where the denominator is the difference between the voltages applied to the + and - inputs. We

use the symbol G for the gain of the complete amplifier with feedback: G ≡ Vout/Vin. This gain

depends on the resistor divider ratio B ≡ R/(R+RF). In feedback amplifier lore A is called the

open loop gain, G is called the closed loop gain, and the product A·B is called the loop gain.

You can see the reason for these names if you imagine opening the feedback loop in Fig. 4.2 by

disconnecting the – input.

V out

V out

A

Figure 4.3 Inverting Amplifier

R

R F

Figure 4.2 Non-inverting Amplifier

Amplifier

+

–

V in

R R F +

–

V in

A

inin

out

VV

VA

Experiment #4 4.4

If we assume that A → ∞, that the op-amp input impedance is infinite, and that the output

impedance is zero, then the behavior of these circuits can be understood using the simple

“Golden Rules”:

I. The voltage difference between the inputs is zero. (“voltage rule”)

II. No current flows into (or out of) each input. (“current rule”)

The “Golden Rule” analysis is very important and is always the first step is designing op-amp

circuits, so be sure you understand it before you read the material below, where we consider the

effect of finite values of A, mainly so that the frequency dependence of the closed loop gain G

can be understood.

GAIN EQUATION – NON-INVERTING CASE

The basic formula for the gain of feedback amplifiers is derived in FC, Section 12.5. From Fig.

4.2 we can see that:

outinout BVVAV

Solving this equation yields:

AB

A

V

VG

in

out

1 (1)

With B=R/(R+RF) the above equation for the closed loop gain then gives

which is the Golden Rule result. If we don't care about corrections due to the finite value of A,

then Equation (1) is not needed and the analysis can be done using just the Golden Rules.

INPUT AND OUTPUT IMPEDANCE – NON-INVERTING CASE

Formulas for the input and output impedance are derived in H&H Section 4.26. The results are

where Ri and Ro are the input and output impedances of the op-amp alone, while the primed

symbols refer to whole the amplifier with feedback. These impedances will be improved from

the values for the bare op-amp if the loop gain A·B is large.

R i Ri(1 AB)

R o Ro / (1 AB)

G 1 R f

R if A 1

Experiment #4 4.5

FREQUENCY DEPENDENCE – NON-INVERTING CASE

The above formulas are still correct when A and/or B depend on frequency. B will be frequency

independent if we have a resistive feedback network, but A always varies with frequency. For

most op-amps, including the LF356 (and others with dominant pole compensation, see H&H

Section 4.34), the open loop gain varies with frequency like an RC low-pass filter:

A A0

1 if

f0

(2)

The 3dB frequency f0 is usually very low, around 10 Hz. Data sheets do not usually give f0

directly; instead they give the dc gain A0 and the unity gain frequency fT, which is the frequency

where the magnitude of the open loop gain A is equal to one. The relation between A0, f0, and

fT is

The frequency dependence of the closed loop gain G can be found by substituting Equation (2)

into Equation (1). You will find the result

The frequency response of the amplifier with feedback is therefore also the same as for an RC

low-pass filter. The 3dB bandwidth fB with feedback is given by

At frequencies well below fB the gain is

GAIN-BANDWIDTH PRODUCT – NON-INVERTING CASE

We can now derive an example of a very important general rule connecting the gain and

bandwidth of feedback amplifiers. Multiplying the low frequency gain G0 by the 3 dB

bandwidth fB gives

In words, this very important formula says that the gain-bandwidth product G0fB equals the unity

fT A0 f0 .

G A0 1 A0B

1 if

f 0 1 A0B

G0

1 if

f B

.

fB f0 1 A0B .

G0 A0

1 A0B.

G0 f B A0

1 A0B f 0 1 A0 B A0 f 0,

G0 fB fT.

Experiment #4 4.6

gain frequency fT. Thus if an op-amp has a unity gain frequency fT of 1 MHz, it can be used to

make a feedback amplifier with a gain of one and a bandwidth of 1 MHz, or with a gain of 10

and a bandwidth of 100 kHz, etc.

GAIN EQUATION – INVERTING CASE

The basic inverting configuration is shown in Figure 4.3. Since the positive input is grounded,

the op-amp will do everything it can to keep the negative input at ground as well. In the limit of

infinite open loop gain the inverting input of the op-amp is a virtual ground, a circuit node that

will stay at ground as long as the circuit is working, even though it is not directly connected to

ground. Since the op-amp inputs draw no current, it follows that

and the dc closed loop gain is

This is the “Golden Rule” result. For the frequency dependence of the gain we need to consider

what happens when A is not infinite, as we did above for the non-inverting case.

The gain equation is discussed in FC 12.4. We use the same definitions as for the non-inverting

case. The op-amp open loop gain is A, and the divider ratio B is given as before by

The voltage

inV at the negative op-amp input is related to Vout and Vin by

)( inoutinin VVBVV

But

inV is also related to Vout by the open-loop gain: Vout = –A

inV . Eliminating

inV from these

two equations gives the gain equation for an inverting amplifier

(3)

(This equation is misprinted on p. 235 of H&H.) The result is almost the same as for a non-

inverting amplifier except for the sign and the factor (1–B) in the numerator. When the open

loop gain is large (A>>1) we have G = 1 – 1/B = – RF/R, the “Golden Rule” result.

INPUT AND OUTPUT IMPEDANCE – INVERTING CASE

Formulas for the input and output impedance for an inverting amplifier are derived in H&H

Section 4.26. When the open loop gain is large, the negative input of the op-amp is a virtual

ground and so the input impedance is just equal to R. This is very different from the non-

inverting case where the input impedance is proportional to A for large A. In practice, the input

Vin R Vout RF

G0 Vout

Vin

RF

R.

B R

R RF

.

G Vout

Vin

A 1 B

1 AB.

Experiment #4 4.7

impedance of an inverting amplifier is not usually greater than about 100 k, while the input

impedance of a non-inverting amplifier can easily be as large as 1012 . When A is not large the

formula for the input impedance is

The formula for the output impedance is the same as for a non-inverting amplifier:

FREQUENCY DEPENDENCE – INVERTING CASE

As for the non-inverting case, all of the above formulas are still correct when A and/or B depend

on frequency. (To make B frequency dependent, R and/or RF may be replaced by networks

containing capacitors or inductors.) We again assume that the op-amp has dominant pole

compensation:

(4)

so that the relation between the open loop 3 dB frequency f0, the dc open loop gain A0 and the

unity gain frequency fT is still

The frequency dependence of the closed loop gain G for the feedback amplifier can be found by

substituting Equation (4) into Equation (3). The result is

When B is frequency independent, the frequency response of the amplifier with feedback is

again the same as an RC low-pass filter. The 3dB bandwidth fB with feedback is the same as for

the non-inverting amplifier

At frequencies well below fB the gain is

The gain-bandwidth product relation is

This is the same (except for the sign) as the non-inverting result when the closed loop gain is

Zin R RF

1 A.

R o Ro / (1 AB).

A A0

1 if

f 0

,

f T A0 f0 .

G

A0 1 B

1 A0 B

1 if

f 0 1 A0 B

G0

1 if

f B

.

f B f 0 1 A0B .

G0 A0 1 B

1 A0B.

G0 f B A0 1 B

1 A0B f 0 1 A0B A0 f 0 1 B ,

G0 f B fT (1 B).

Experiment #4 4.8

large (B << 1, G0 >> 1), but at unity closed loop gain (B = 1/2, G0 = -1) the inverting

amplifier has only half as much bandwidth as a non-inverting follower.

Experiment #4 4.9

Problems

1. (2 points) Data for the LF356 op-amp (same as LF156) are given at the National

Semiconductor Web Site: http://www.national.com/ds/LF/LF155.pdf (there is a link

from the 3330 web page and a copy in the lab). Verify that the following values are

reasonable approximations to use in solving the problems. Identify your source of each

item by quoting the graph or table number and page number.

DC voltage gain A0 = 2·105 (same as AVOL in tables)

Unity gain frequency fT = 5 MHz (same as GBW in tables)

Maximum output voltage Vsat = ± 13 V (same as VO in tables)

Maximum slew rate SR=(dVout/dt)max = 12 Volts/µsec

Maximum output current (IO)max = 25 mA

Input resistance Ri = 1012 (RIN in tables)

Output resistance Ro = 40 (See the theory section for help with Ro.)

Input offset voltage VOS = 3 mV

Input bias current IB = 30pA

2. (3 points) Calculate the values of low frequency gain G0 (choose the correct version of

the formula) and the bandwidth fB for the non-inverting amplifier in Fig. 4.2. Consider

the cases with RF = 100 k for all, and R = 100 , 1 k, and 10 k. Also consider the

voltage follower, with RF=0 and R= (note there are four cases). Draw Bode plots for

the open loop gain and the four closed loop gains on the same graph. Estimate the input

and output impedances of the amplifier with RF = 10 k and R = 100 for 1 kHz sine

waves.

3. (3 points) What is the maximum output amplitude for 5 MHz sine waves if they are not to

be distorted by the slew rate of the LF356 op-amp? (For help, see H&H p. 192.) Draw a

voltage vs. time plot showing a sine wave input and a typical slew rate limited output on

the same plot. (Note that an amplifier limited by slew rate will change a curved line to a

straight line on the oscilloscope.)

4. (2 points) Design an inverting amplifier (Fig. 4.3) with a closed-loop gain somewhere

between 10 and 100. Select appropriate values for R and RF. Choose RF large enough so

that the maximum output current of the op-amp is not exceeded when the output is near

the saturation voltage. Predict the bandwidth and the input impedance at the signal input.

How would you measure the input impedance in the lab? [The DMM is not going to

properly read the input or output impedance when connected to a circuit because of all

the other resistors and bias voltages present. Here is a hint: Recall that when the signal

generator with a 50 output impedance has a 50 load the signal into the load drops by

a factor of two.]

Experiment #4 4.10

Apparatus

This experiment will use both +15V and -15V to power the LF356 op-amp. Look back at

Experiment 2 to remember how to connect the power supply to your prototyping board.

Turn off the power while wiring your op-amp circuits. Figure 4.4 shows pin-out data and a

layout for the gain=100 amplifier.

0 V

Figure 4.4 Op-amp pin diagram and possible layout for gain 100 amplifier

+

–15 V

+15 V

Vn

in

Vut

out

R F

R +

–15 V

356

+

–

local ground

panel ground

356

+

–

+15 V

Input Vut

o

2

4 3

6 7

7

8

6

5 4

3

1

2

+

– bal. 1

V

V

–15 V

NC

+15 V

V

bal. 2

–

+ out

1

8

4

5

+ –

view from above

0 V

–15 V

+15 V

0 V

+

+

R

R

F

Vut

out

Vn

in

bypass cap

Everyone makes mistakes in wiring-up circuits. Thus, it’s a good idea to check your circuit

before applying power. You may well save a transistor or chip from burnout and save yourself a

lot of frustration.

The following procedure will help you wire up a circuit accurately:

1. Draw a complete schematic in your lab book, including all ground and power

connections, and all IC pin numbers. Try to layout your prototype so the parts are

arranged in the same way as on the schematic, as far as possible.

2. Know the color or number code for resistors and capacitors so you can avoid putting the

wrong part into the circuit. When in doubt about the value of a part, measure it with an

Ohm meter or impedance bridge.

3. Adhere to a color code for wires. Red for positive power and black for ground is a very

widely honored standard. Beyond that, just try to be consistent. For example:

0 V (ground) Black

+15 V Red

-15 V Blue

analog signals Yellow

digital signals Purple

Experiment #4 4.11

The op-amp chip sits across a groove in the circuit board. Before inserting a chip, gently

straighten the pins. After insertion, check visually that no pin is broken or bent under the chip.

To remove the chip, use a small screwdriver in the groove to pry it out.

You will have less trouble with spontaneous oscillations if the circuit layout is neat and compact,

especially the feedback path should be as short as possible to reduced unwanted capacitive

coupling and lead inductance.

To help prevent spontaneous oscillations due to unintended coupling via the power supplies, use

bypass capacitors to filter the supply lines. A bypass capacitor between each power supply lead

and ground, will provide a miniature current “reservoir” that can quickly supply current when

needed. This capacitor is normally in the range 1 F – 10 F. Compact capacitors in this range

are usually electrolytic tantalum or aluminum and are polarized, meaning that one terminal

(marked +) must always be positive relative to the other. If you put a polarized capacitor in

backwards, it will burn out. It is often helpful to put a smaller (0.01-0.1 µF) non-polar ceramic

capacitor between power and ground as well. Ceramic capacitors offer less capacitance than

electrolytic ones, but work better at high frequencies. Bypass capacitors should be placed as

close as possible to the op-amp supply pins.

Experiment

GETTING STARTED: TESTING THE OP-AMP

You can save yourself some frustration by testing your

op-amp chips to make sure they are not burned out.

Connect the op-amp as a voltage follower with the input

grounded (see Fig. 4.5). Since Vout = Vin for a follower,

you expect Vout = 0 for this circuit. A "bad" chip will

often have Vout = ±Vsat because of a burned-out output

circuit.

Measure the DC voltages with a multimeter. Be careful

not to connect pins 6 & 7, since this will burn out the op-amp. You should find Pin 4 = –15 V,

Pin 7 = +15 V, Pin 2 = 0 V, Pin 3 = 0 V, Pin 6 = 0 V. The small voltage on pin 2 is equal to the

input offset voltage VOS, typically less than 3 mV. If Vout = ± 15 V, then the chip is bad.

Throw it in the trash to save another person from having to deal with it. (In case you are

wondering, the LF356 costs $0.50 when purchased in quantities of 100. Check out a distributor

like www.digikey.com if you are curious about the cost of any part we use.)

0 V

Figure 4.5 Test circuit for op-amp

+

+15 V

Vut

o

+

–15 V

356

+

–

local ground

panel ground

2

6

7

4 3

Experiment #4 4.12

1. VOLTAGE FOLLOWER

The following section makes use of the basic test configuration as shown in Figure 4.6:

Counter/

Timer

Function

Generator

Circuit

Board

V out V in

V in

Trigger

Trig.

Out

Sig.

Out

Oscilloscope

Channel 1

Channel 2

Channel 4

Figure 4.6 Test Set-up

DC Power

Supply

0 +15 –15

This is similar to the one we used for Experiment #3. The Counter-timer is optional in this set-

up, since the function generator has a direct frequency readout that can also be verified using the

measurement functions on the TDS3014B oscilloscope.

The voltage follower has no voltage gain (G0=1), but it lets you convert a signal with high

impedance (i.e. very little current) to a much lower impedance output for driving loads. The

voltage follower is also often called a unity gain buffer.

The voltage follower circuit is nearly the same as

the test circuit, except that now a signal enters the

positive input (Figure 4.7).

Set your function generator to produce a 1 kHz

sine wave (~1 volt p-p initially). Then:

1A. Measure the low frequency gain G0 by

measuring Vin and Vout with the voltage cursors.

Make both waveforms at least 4 divisions high for

accuracy. Compare your results with the

prediction.

1B. Now vary the frequency and look for

0 V

Figure 4.7 Voltage Follower

+

5

6

–

356

–15 V

+15 V

Vn

in

V out

+

+

2

3 4 6

7

Experiment #4 4.13

deviation from the performance of an ideal follower. The measurement at high frequency will

depend on many details of your setup and you are unlikely to find a simple RC filter type

falloff. Using the 10X probe, measure the gain at every decade in frequency from 10 MHz down

to 10 Hz. Do you find any deviation from unity gain? Be sure that the output amplitude is below

the level affected by the slew rate! Plot the low and high frequency data on a Bode diagram. Do

you find a simple fall-off as suggested by the theory for the ideal follower (fT=fB)? If so, find the

3dB frequency. How does your measured 3dB frequency compare to what is expected from the

op-amp datasheet? Even if the fall-off doesn't have a simple behavior, at what frequency do you

observe more than a 10% deviation from the ideal gain?

If you observed "ideal" behavior, you're lucky! At frequencies above a few MHz, the simple

model of the frequency response of the op-amp (see theory section) is not accurate. Furthermore,

once you are in this frequency range, many physical details of your circuit and breadboard can

have large effects in the circuit. Building reliable circuits at these frequencies typically requires

careful attention to grounding and minimization of capacitive and inductive coupling between

circuit elements and to ground. At lower frequencies, our op-amp model will work much better.

2. GAIN = 100 NON-INVERTING AMPLIFIER

Change the negative feedback loop to the one in Figure 4.8, with RF = 10 k and R = 100 .

Use 1%, 1/2 Watt, precision resistors for the feedback loop. Measure R and RF with the

multimeter before inserting them into the circuit board. Recalculate G0 and fB from these

measured values and the op-amp's value of fT measured in the last part.

2A. Measure the low frequency gain G0 for 1 kHz sine waves. Vary the input amplitude until

you observe saturation in the output. What are the output saturation levels, +Vsat and -Vsat?

Now, reduce the input amplitude so that the output voltage is less than half the saturated value.

2B. Determine the 3dB bandwidth fB. Then vary

the frequency to obtain data at decade intervals for

checking your Bode diagram.

2C. Using the gain-bandwidth relation G0fB=fT,

determine the fT for your op-amp. How does this

compare with the value from the op-amp datasheet?

2D. The input impedance of the amplifier should be

exceptionally high. Can you see any change in

output when you connect a 1 M resistor in series

0 V

Figure 4.8 Gain of 100 Amplifier

+

–

356

–15 V

+15 V

V in

V out

+

+

2

3 4 6

7

R R F

Experiment #4 4.14

with the input? Is this consistent with your homework result?

2E. The output impedance should be very low, provided that the output current is less than the

maximum available (25 mA). What happens when you connect a load of 220 from output to

ground? Explore as a function of output amplitude. Be sure to pull out this 220 before

proceeding to the next section.

3. INVERTING AMPLIFIER

Build the inverting amplifier that you designed in Problem 5.

3A. Measure the gain versus frequency and record the result on a Bode plot, comparing with

your prediction.

3B. Find the gain-bandwidth product and see how it compares to the value your observed for the

gain 100 non-inverting amplifier.

3C. Measure the input impedance and see if it agrees with your expectations.